Dipartimento di Matematica Alma Mater Studiorum ...simoncin/numoc.pdf · Dipartimento di Matematica...

Order reduction numerical methods for the

algebraic Riccati equation

V. Simoncini

Dipartimento di Matematica

Alma Mater Studiorum - Universita di Bologna

[email protected]

1

The problem

Find X ∈ Rn×n such that

AX+XA⊤ −XBB

⊤X+ C

⊤C = 0

with A ∈ Rn×n, B ∈ R

n×p, C ∈ Rs×n, p, s = O(1)

Rich literature on analysis, applications and numerics:

Lancaster-Rodman 1995, Bini-Iannazzo-Meini 2012, Mehrmann etal 2003 ...

We focus on the large scale case: n ≫ 1000

Different strategies

(Inexact) Kleinman iteration (Newton-type method)

Projection methods

Invariant subspace iteration

(Sparse) multilevel methods

....

2

The problem

Find X ∈ Rn×n such that

AX+XA⊤ −XBB

⊤X+ C

⊤C = 0

with A ∈ Rn×n, B ∈ R

n×p, C ∈ Rs×n, p, s = O(1)

Rich literature on analysis, applications and numerics:

Lancaster-Rodman 1995, Bini-Iannazzo-Meini 2012, Mehrmann etal 2003 ...

We focus on the large scale case: n ≫ 1000

Different strategies

• (Inexact) Kleinman iteration (Newton-type method)

• Projection methods

• Invariant subspace iteration

• (Sparse) multilevel methods

• ....

3

Newton-Kleinman iteration

Assume A stable. Compute sequence {Xk} with Xk →k→∞ X

(A−XkBB⊤)Xk+1 +Xk+1(A

⊤ −BB⊤Xk) + C

⊤C +XkBB

⊤Xk = 0

1: Given X0 ∈ Rn×n such that X0 = X⊤

0 , A⊤ −BB⊤X0 is stable

2: For k = 0, 1, . . . , until convergence

3: Set A⊤

k = A⊤ −BB⊤Xk

4: Set C⊤

k = [XkB, C⊤]

5: Solve AkXk+1 +Xk+1A⊤

k + C⊤

k Ck = 0

Critical issues:

The full matrix Xk cannot be stored (sparse or low-rank approx)

Need a computable stopping criterion

Each iteration k requires the solution of the Lyapunov equation:

(Benner, Feitzinger, Hylla, Saak, Sachs, ...)

4




⊤ −BB⊤Xk) + C

⊤C +XkBB

⊤Xk = 0


0 , A⊤ −BB⊤X0 is stable


3: Set A⊤

k = A⊤ −BB⊤Xk

4: Set C⊤

k = [XkB, C⊤]


k + C⊤

k Ck = 0

Critical issues:

The full matrix Xk cannot be stored (sparse or low-rank approx)

Need a computable stopping criterion

Each iteration k requires the solution of the Lyapunov equation:


5




⊤ −BB⊤Xk) + C

⊤C +XkBB

⊤Xk = 0


0 , A⊤ −BB⊤X0 is stable.


3: Set A⊤

k = A⊤ −BB⊤Xk

4: Set C⊤

k = [XkB, C⊤]


k + C⊤

k Ck = 0

Critical issues:

• The full matrix Xk cannot be stored (sparse or low-rank approx)

• Need a computable stopping criterion

• Each iteration k requires the solution of the Lyapunov equation


6

Galerkin projection method for the Riccati equation

Given the orth basis Vk for an approximation space, determine

Xk = VkYkV⊤

k

to the Riccati solution matrix by orthogonal projection:

Galerkin condition: Residual orthogonal to approximation space

V⊤

k (AXk +XkA⊤ −XkBB

⊤Xk + C

⊤C)Vk = 0

giving the reduced Riccati equation

(V ⊤

k AVk)Y +Y(V ⊤

k A⊤Vk)−Y(V ⊤

k BB⊤Vk)Y + (V ⊤

k C⊤)(CVk) = 0

Yk is the stabilizing solution (Heyouni-Jbilou 2009)

Key questions:

Which approximation space?

Is this meaningful from a Control Theory perspective?

7



Xk = VkYkV⊤

k



V⊤


⊤Xk + C

⊤C)Vk = 0


(V ⊤

k AVk)Y +Y(V ⊤

k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0


Key questions:



8



Xk = VkYkV⊤

k



V⊤


⊤Xk + C

⊤C)Vk = 0


(V ⊤

k AVk)Y +Y(V ⊤

k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0


Key questions:



9



Xk = VkYkV⊤

k



V⊤


⊤Xk + C

⊤C)Vk = 0


(V ⊤

k AVk)Y +Y(V ⊤

k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0


Key questions:

• Which approximation space?

• Is this meaningful from a Control Theory perspective?

10

Dynamical systems and the Riccati equation

AX+XA⊤ −XBB⊤X+ C⊤C = 0

Time-invariant linear system

x(t) = Ax(t) +Bu(t), x(0) = x0

y(t) = Cx(t),

u(t) : control (input) vector; y(t) : output vector

x(t) : state vector; x0 : initial state

Minimization problem for a Cost functional: (simplified form)

infu

J (u, x0) J (u, x0) :=

∫ ∞

0

(x(t)⊤C⊤Cx(t) + u(t)⊤u(t))dt

11



Time-invariant linear system

x(t) = Ax(t) +Bu(t), x(0) = x0

y(t) = Cx(t),

u(t) : control (input) vector; y(t) : output vector

x(t) : state vector; x0 : initial state

Minimization problem for a Cost functional: (simplified form)

infu

J (u, x0) J (u, x0) :=

∫ ∞

0

(x(t)⊤C⊤Cx(t) + u(t)⊤u(t)

)dt

12



infu

J (u, x0) J (u, x0) :=

∫ ∞

0

(x(t)⊤C⊤Cx(t) + u(t)⊤u(t)

)dt

theorem Let the pair (A,B) be stabilizable and (C,A) observable.

Then there is a unique solution X ≥ 0 of the Riccati equation. More-

over,

i) For each x0 there is a unique optimal control, and it is given by

u∗(t) = −B⊤X exp((A−BB⊤

X)t)x0 for t ≥ 0

ii) J (u∗, x0) = x⊤0 Xx0 for all x0 ∈ Rn

see, e.g., Lancaster & Rodman, 1995

13

Order reduction of dynamical systems by projection

Let Vk ∈ Rn×dk have orthonormal columns, dk ≪ n

Let Tk = V ⊤k AVk, Bk = V ⊤

k B, C⊤k = V ⊤

k C⊤

Reduced order dynamical system:

˙x(t) = Tkx(t) +Bku(t), x(0) = x0 :=V ⊤k x0

y(t) = Ckx(t)

xk(t) = Vkx(t) ≈ x(t)

Typical frameworks:

• Transfer function approximation

• Model reduction

14

The role of the projected Riccati equation

Consider again the reduced Riccati equation:

(V ⊤k AVk)Y +Y(V ⊤

k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0

that is

TkY +YT⊤k −YBkB

⊤k Y + C⊤

k Ck = 0 (∗)

Theorem. Let the pair (Tk, Bk) be stabilizable and (Ck, Tk) observ-

able. Then there is a unique solution Yk ≥ 0 of (*) that for each x0

gives the feedback optimal control

u∗(t) = −B∗kYk exp((Tk −BkB

∗kYk)t)x0, t ≥ 0

for the reduced system.

If there exists a matrix K such that A−BK is passive, then the pair

(Tk, Bk) is stabilizable.

15




k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0

that is

TkY +YT⊤k −YBkB

⊤k Y + C⊤

k Ck = 0 (∗)





∗kYk)t)x0, t ≥ 0


If there exists a matrix K such that A−BK is passive, then the pair

(Tk, Bk) is stabilizable.

16




k A⊤Vk)−Y(V ⊤


k C⊤)(CVk) = 0

that is

TkY +YT⊤k −YBkB

⊤k Y + C⊤

k Ck = 0 (∗)





∗kYk)t)x0, t ≥ 0


♣ If there exists a matrix K such that A−BK is passive, then the

pair (Tk, Bk) is stabilizable.

17

Projected optimal control vs approximate control

⋆ Our projected optimal control function:

u∗(t) = −B⊤k Yk exp((Tk −BkB

⊤k Yk)t)x0, t ≥ 0

with Xk = VkYkV⊤k

⋆ Commonly used approximate control function:

If X is some approximation to X, then

u(t) := −B⊤Xx(t)

where x(t) := exp((A−BB⊤X)t)x0

u∗ 6= u

They induce different actions on the functional J , even for X = Xk

18

Projected optimal control vs approximate control

Xk = VkYkV⊤k

Residual matrix: Rk := AXk +XkA−XkBB⊤Xk + C⊤C

⋆ Projected optimal control function:

u∗(t) = −B⊤k Yk exp((Tk −BkB

⊤k Yk)t)

theorem. Assume that A − BB⊤Xk is stable and that u(t) :=

−B⊤Xkx(t) approx control. Then

|J (u, x0)− Jk(u∗, x0)| = Ek, with Ek ≤ ‖Rk‖2α

x⊤0 x0,

where α > 0 is such that ‖e(A−BB⊤Xk)t‖ ≤ e−αt for all t ≥ 0.

Note: |J (u, x0)− Jk(u∗, x0)| is nonzero for Rk 6= 0

19

On the choice of approximation space

Approximate solution Xk = VkYkV⊤

k , with

(V ⊤

k AVk)Yk +Yk(V⊤

k A⊤Vk)−Yk(V

⊤

k BB⊤Vk)Yk + (V ⊤

k C⊤)(CVk) = 0

Krylov-type subspaces: (extensively used in the linear case)

• Kk(A,C⊤) := Range([C⊤, AC⊤, . . . , Ak−1C⊤]) (Polynomial)

• EKk(A,C⊤) := Kk(A,C⊤) +Kk(A−1, A−1C⊤) (EKSM, Rational)

• RKk(A,C⊤, s) :=

Range([C⊤, (A− s2I)−1C⊤, . . . ,

k−1∏

j=1

(A− sj+1I)−1

C⊤])

(RKSM, Rational)

⋆ Matrix BB⊤ not involved

⋆ Parameters sj (adaptively) chosen in field of values of −A

20



k , with

(V ⊤

k AVk)Yk +Yk(V⊤

k A⊤Vk)−Yk(V

⊤


k C⊤)(CVk) = 0




• RKk(A,C⊤, s) :=

Range([C⊤, (A− s2I)−1C⊤, . . . ,

k−1∏

j=1

(A− sj+1I)−1

C⊤])

(RKSM, Rational)

⋆ Matrix BB⊤ not involved (nonlinear term!)


21



k , with

(V ⊤

k AVk)Yk +Yk(V⊤

k A⊤Vk)−Yk(V

⊤


k C⊤)(CVk) = 0




• RKk(A,C⊤, s) :=

Range([C⊤, (A− s2I)−1C⊤, . . . ,

k−1∏

j=1

(A− sj+1I)−1

C⊤])

(RKSM, Rational)

⋆ Matrix BB⊤ not involved (nonlinear term!)


22

Performance of solvers

Problem: A: 3D Laplace operator, B, C randn matrices, tol=10−8

(n, p, s) = (125000, 5, 5)

its inner its time space dim rank(Xf )

Newton X0 = 0 15 5, . . . , 5 808 100 95

GP-EKSM 20 531 200 105

GP-RKSM 25 524 125 105

(n, p, s) = (125000, 20, 20)

its inner its time space dim rank(Xf )

Newton X0 = 0 19 5, . . . , 5 2332 400 346

GP-EKSM 15 622 600 364

GP-RKSM 20 720 400 358

GP=Galerkin projection

(V.Simoncini & D.Szyld & M.Monsalve, 2014)

23

A numerical example on the role of BB⊤

Consider the 500× 500 Toeplitz matrix

A = toeplitz(−1, 2.5, 1, 1, 1), C = [1,−2, 1,−2, . . .], B = 1

0 5 10 15 20 25 30 35 4010

−12

10−10

10−8

10−6

10−4

10−2

100

102

104

space dimension

ab

so

lute

re

sid

ua

l n

orm

RKSM

Parameter computation:

Left: adaptive RKSM on A Right: adaptive RKSM on A−BB⊤Xk

(Lin & Simoncini 2015)

24

A numerical example on the role of BB⊤

Consider the 500× 500 Toeplitz matrix

A = toeplitz(−1, 2.5, 1, 1, 1), C = [1,−2, 1,−2, . . .], B = 1

0 5 10 15 20 25 30 35 4010

−12

10−10

10−8

10−6

10−4

10−2

100

102

104

space dimension

ab

so

lute

re

sid

ua

l n

orm

RKSM

0 5 10 15 20 25 30 35 4010

−12

10−10

10−8

10−6

10−4

10−2

100

102

104

space dimension

ab

so

lute

re

sid

ua

l n

orm

RKSM

Parameter computation:

Left: adaptive RKSM on A Right: adaptive RKSM on A−BB⊤Xk

(Lin & Simoncini 2015)

25

On the residual matrix and adaptive RKSM

Rk := AXk +XkA−XkBB⊤Xk + C⊤C

theorem. Let Tk = Tk −BkB⊤k Yk. Then

Rk = RkV⊤k + VkR

⊤k , with Rk = AVkYk + VkYkT ⊤

k + C⊤(CVk)

so that ‖Rk‖F =√2‖Rk‖F

At least formally:

⇒ VkYkV⊤k is a solution to the Riccati equation (Rk = 0) if and only

if Zk = VkYk is the solution to the Sylvester equation (Rk = 0)

26

On the residual matrix and adaptive RKSM

Rk = RkV⊤k + VkR

⊤k

Expression for the semi-residual Rk:

theorem. Assume C⊤ ∈ Rn, Range(Vk)= RKk(A,C

⊤, s). Assume

that Tk = Tk −BkB⊤k Yk is diagonalizable. Then

Rk = ψk,Tk(A)C⊤CVk(ψk,Tk

(−T ⊤k ))−1.

where

ψk,Tk(z) =

det(zI − Tk)∏kj=1(z − sj)

(see also Beckermann 2011 for the linear case)

27

On the choice of the next parameters sk+1

Rk = ψk,Tk(A)C⊤CVk(ψk,Tk

(−T ⊤k ))−1.

with ψk,Tk(z) = det(zI−Tk)∏

kj=1

(z−sj)

⋆ Greedy strategy: Next shift should make (ψk,Tk(−T ⊤

k ))−1 smaller

⇓

Determine for which s in the spectral region of Tk the quantity

(ψk,Tk(−s))−1 is large, and add a root there

sk+1 = arg maxs∈∂Sk

∣∣∣∣1

ψk,Tk(s)

∣∣∣∣

Sk region enclosing the eigenvalues of −Tk = −(Tk −BkB⊤k Yk)

(This argument is new also for linear equations)

28

Selection of sk+1 in RKSM. An example

A: 900× 900 2D Laplacian, B = t1 with tj = 5 · 10−j ,

C = [1,−2, 1,−2, 1,−2, ...]

0 5 10 15 20 2510

−10

10−8

10−6

10−4

10−2

100

space dimension

ab

so

lute

re

sid

ua

l n

orm

t3

t3

t2

t2

t1

t1

RKSM convergence with and without modified shift selection as t varies

Solid curves: use of Tk Dashed curves: use of Tk

29

Further results not presented but relevant

• Stabilization properties of the approx solution Xk

• Accuracy tracking as the approximation space grows

• Interpretation via invariant subspace approximation

(V.Simoncini, 2016)

30

Wrap-up and outlook

♥ Projection-type methods fill the gap between MOR and Riccati equation

♥ Clearer role of the non-linear term during the projection

Projected Differential Riccati equations

(see, e.g., Koskela & Mena, tr 2017)

Parameterized Algebraic Riccation equations

(see, e.g., Schmidt & Haasdonk, tr 2017)

References

- V. Simoncini, Daniel B. Szyld and Marlliny Monsalve,

On two numerical methods for the solution of large-scale algebraic Riccati

equations IMA Journal of Numerical Analysis, (2014)

- Yiding Lin and V. Simoncini,

A new subspace iteration method for the algebraic Riccati equation

Numerical Linear Algebra w/Appl., (2015)

- V.Simoncini, Analysis of the rational Krylov subspace projection method for

large-scale algebraic Riccati equations SIAM J. Matrix Anal.Appl, (2016)

- V. Simoncini, Computational methods for linear matrix equations SIAM Review,

(2016).

31

Wrap-up and outlook



♠ Projected Differential Riccati equations


♠ Parameterized Algebraic Riccation equations


References








large-scale algebraic Riccati equations SIAM J. Matrix Analysis Appl, (2016).

- V. Simoncini, Computational methods for linear matrix equations (Survey article)

SIAM Review, (2016).

32

Wrap-up and outlook



♠ Projected Differential Riccati equations


♠ Parameterized Algebraic Riccation equations


References








large-scale algebraic Riccati equations SIAM J. Matrix Anal.Appl, (2016)

- V. Simoncini, Computational methods for linear matrix equations SIAM Review,

(2016)

33

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